Some identities on Catalan numbers and hypergeometric functions via Catalan matrix power (Q547979)

The author defines the Pascal matrix \(P_{n}(x)\) to be the \(n\times n\) upper triangular matrix whose \((i,j)\)th entry is \(x^{i-j}\binom{i-1}{j-1}\) and the Catalan matrix \(C_{n}(x)\) as the \(n\times n\) triangular matrix whose \((i,j)\)th entry is \(x^{i-j}\frac{1}{i-j+1}\binom{2i-2j}{i-j}\). He then defines \(G_{n}(x):=C_{n}(x)^{-1}P_{n}(x)\) where the entries of \(G_{n}(x)\) can be described explicitly in terms of the binomial coefficients and the hypergeometric function \(_{2}F_{1}\). Using various products of these matrices the author derives a series of identities involving the Catalan numbers and values of \(_{2}F_{1}\) and \(_{3}F_{2}\).